The context in which cycle matrices were discovered and
some of their basic symmetry properties are explained in Gerdes (2002a
and 2005). The book Adventures in the World of Matrices
(Gerdes 2004) presents an introduction to alternating
cycle matrices and cycle matrices of any period p. The papers Gerdes
(2002b and c) introduce helix and cylinder
matrices and explore some of their relationships with cycle matrices.

In the following paper some further symmetry properties
of alternating cycle matrices will be presented. The paper concludes with
a geometrical interpretation of a class of permutation matrices that are
simultaneously alternating cycle matrices. It is hoped that the article
may underscore some of the beautiful symmetric properties of cycle matrices
and may stimulate further exploration.

Definition and generalities

Alternating cycle matrices constitute a particular class
of square matrices. If m is an odd number (m = 2n+1),
a matrix of dimensions m × m is called a positive alternating
cycle matrix if all the elements on its principal diagonal are equal and
if it is composed of n cycles of alternating numbers. Figure
1 illustrates the structure and general form of a positive alternating
cycle matrix of dimensions 5×5.

Structure and general form of a positive alternating cycle
matrix of dimensions 5×5

Figure 1

If m is an odd number (m = 2n+1),
a matrix of dimensions
m × m is called a negative alternating
cycle matrix if all the elements on its secondary diagonal are equal and
if it is composed of n cycles of alternating numbers. Figure
2 illustrates the structure and general form of a negative alternating
cycle matrix of dimensions 5×5.

Structure and general form of a negative alternating cycle
matrix of dimensions 5×5

Figure 2

If m is an even number (m = 2n),
a matrix of dimensions
m × m is called a positive alternating
cycle matrix if each of its diagonals is constant and if it is furthermore
composed of n-1 cycles of alternating numbers. Figure 3 illustrates
the structure and general form of a positive alternating cycle matrix of
dimensions 6×6.

Structure and general form of a positive alternating cycle
matrix of dimensions 6×6

Figure 3

If m is an even number (m = 2n),
a matrix of dimensions
m × m is called a negative alternating
cycle matrix if it is composed of n cycles of alternating numbers. Figure
4 illustrates the structure and general form of a negative alternating
cycle matrix of dimensions 6×6.

Structure and general form of a negative alternating cycle
matrix of dimensions 6×6

Figure 4

The adjectives ‘positive’ and ‘negative’ are justified
by the fact that the multiplication of positive and negative alternating
cycles satisfies the same multiplication rules as positive and negative
numbers: + × + = +, + × – = –, – ×
+ = –, and – × – = +. Moreover, if A and B are
two positive alternating cycle matrices of the same dimensions, then AB
= BA. If C and D are two negative alternating cycle
matrices of the same dimensions, then CD and DC are mutually
symmetric: DC is the transposed matrix of CD.

In the following some further symmetries of alternating
cycle matrices will be presented.

Inverse matrices

Theorem 1

The inverse matrices of non-singular
alternating cycle matrices are also alternating cycle matrices.

Proof

Let A be a non-singular positive alternating cycle
matrix of dimensions m×m. Let r = (b1,
…, bm) be the first row of B = A-1.
We may construct the unique positive alternating cycle matrix C
that has
r as its first row. On its turn CA is a positive
alternating cycle matrix, being its first row equal to the first row of
BA.
As B = A-1, we have BA = A-1A
= I, where I represents the identity matrix of dimensions
m×m. The first row of BA is (1, 0, …, 0). By consequence,
the first row of CA is (1, 0, …, 0). As CA is a positive
alternating cycle matrix, its principal diagonal is composed of 1’s and
its cycles composed of 0’s. It follows that CA = I, and thus C
is the inverse matrix of A: C = B = A-1. As C
is by construction a positive alternating cycle matrix, we have proven
that the inverse matrix of a non-singular positive alternating cycle matrix
is a positive alternating cycle matrix itself.

The proof for the case that A is a non-singular
negative alternating cycle matrix is analogous to the proof given.

Determinants

Theorem 2

Let P be any positive alternating cycle matrix of
dimensions m × m (m = 3,
4,
5, ….). Let N be the negative alternating cycle matrix that has
the same first row as P. For the determinants of P and N, we have

det P = (-1)ndet N,

where n is given by m = 2n+2 if m
is an even number and by m = 2n+1 if m is an odd number.Proof

By interchanging the 2nd and the 3rd
row, the 4th and the 5th row, …, and the (2n)th
and the (2n+1)th row, matrix P is transformed
into matrix N. Interchanging two rows of a matrix implies changing
the sign of the corresponding determinant. As we have a total of n
interchanges, it follows that

Two (positive or negative) alternating cycle matrices
A
and B may be called equivalent, if each row of A is
a row of B, and vice versa.

By interchanging several pairs of rows, matrix A
can be transformed into any equivalent matrix B. By consequence,
we have

Theorem 3

If A and B are equivalent alternating cycle matrices,
then |det A| = |det B|.

Equivalent alternating cycle matrices

Let A be a positive alternating cycle matrix of
dimensions
m×m. If we construct another positive
alternating cycle matrix B that has the first row of A as
one of its rows, what can be said about B?

Let us consider an example. F is a positive alternating
cycle matrix of dimensions 5×5. We construct a positive alternating
cycle matrix G such that the first row of F becomes the fourth
row of G (see Figure 6).

Transformation of matrix F in matrix G

Figure 6

In this example we see that all rows of F appear
as rows of G, and vice versa. The same holds for the columns
of F and G. How can we explain this phenomenon?

Let V4 be the positive alternating cycle
matrix of dimensions 5×5 that has 0’s on the principal diagonal,
one cycle of 0’s and one alternating cycle of 0’s and 1’s such that the
first element of its 4th row is 1 (Figure 7).
V4
is a permutation matrix, i.e. a square matrix whose elements in any row
(or any column) are all zero, except for one element equal to unity.

Permutation matrix V4

Figure 7

Let us multiply the matrices V4 and
F.
As the 4th row of V4 is (1, 0, 0, 0, 0), the
4th row of V4F is (a, b,
c,
d, e), that is the first row of F. As V4
and F are positive alternating cycle matrices, the product is also
a positive alternating cycle matrix. There exists only one positive alternating
cycle matrix that has (a, b, c, d, e)
as its 4th row. It follows that V4F =
G.
Similarly FV4 = G. We may conclude that
F
and G are equivalent.

Based on this and similar experiences we may conjecture
the following generalisation:

Theorem 4

Let A be a positive alternating cycle matrix of
dimensions m × m. If B is defined as the positive alternating
cycle matrix that has the first row of A as its pth row (p =
2, …, m), then A and B are equivalent
alternating cycle matrices.

Proof

Let Vp be the positive alternating cycle
matrix of dimensions m× m whose elements in the first
column are all zero, except for the first element of the pth
row that is equal to unity. By consequence, Vp is a permutation
matrix and its pth row is (1, 0, …, 0). Vp
A is a positive alternating cycle matrix and its pth
row is equal to the first row of matrix A. As there exists only
one positive alternating cycle matrix that has the first row of A
as its pth row, it follows immediately that Vp
A = B, and all rows of B are rows of A. We conclude
that A and B are equivalent alternating cycle matrices.

Before considering in more detail the properties of the
particular permutation matrices Vp, let us analyse the
relationship between the inverse matrices of A and B in the
case that A and B are non-singular.

We may conjecture

Theorem 5

If A and B are equivalent non-singular alternating
cycle matrices, then A-1 and B-1 are
equivalent alternating cycle matrices.

Proof

We have A Vp= Vp A
= B, for a certain p = 2, …, m. The inverse of Vp
is another matrix of the same type Vq(cf. Gerdes
2005). We have

The matrices Vpconstitute
a particular class of permutation matrices: the permutation matrices that
are simultaneously positive alternating cycle matrices.

Let us observe the example V4 when m
= 5 (Figure 7). Multiplying a positive alternating cycle
matrix of dimensions 5×5 by V4, transforms
the first row of A into the 4th row of the new matrix
B
(= A V4), the 4th row of A becomes
the 3rd of B, the 3rd row of A becomes
the 2nd of of B, the 2nd row of A becomes
the 5th of B, and the 5th row of A
becomes the 1st of B. In other words, the cycle
notation for the permutation of the rows is (1 4 3 2 5).

How can we understand and interpret the particular sequence
of the numbers 1 to 5 in the cycle notation of these permutations?

In general, any cycle of a cycle matrix passes exactly
twice through each row. In the particular case of V4,
the non-zero cycle passes exactly twice to each of the five rows. Correspondingly,
let us mark 10 dots at equal intervals around a circle and number these
dots twice, on the left and on the right, from 1 to 5, as illustrated in
Figure
8.

Marking and numbering of ten dots

Figure 8

The permutation (1 4 3 2 5) can now be represented by
a polygon: starting with the 1-dot on the left side, advancing clockwise
four dots one arrives at the 4-dot on the right side (draw the corresponding
segment 1-4); advancing another four dots one arrives at 3-dot on the left
side (draw the corresponding segment 4-3); advancing once more four dots,
one arrives at the 2-dot on the right (segment 3-2), and then
at the 5-dot on the left (drawing the segment 2-5) and finally by advancing
four more dots one returns to the initial 1-dot on the left (Figure
9). The final representation of the permutation (1 4 3 2 5) is that
of a regular pentagram.

Regular pentagram corresponding to permutation (1 4 3
2 5)

Figure 9

The inverse permutation of (1 4 3 2 5) is (5 2 3 4 1)
that may be written as (1 5 2 3 4), and its representation is the vertical
mirror image of the first (Figure 10).

Polygonal-circle representation of the permutation (1
5 2 3 4)

Figure 10

On the one hand, the mutually inverse permutations (1
4 3 2 5) and (1 5 2 3 4) correspond to the positive alternating cycle matrices
V4
and V5 with V4V5
= V5V4 = I. At the same time
these matrices are mutually symmetrical: a reflection in the principal
diagonal of V4 transforms V4 into V5,
and vice versa (Figure 11). In other words,
V5 is the
transposed matrix of V4:
V5 = (V4)T.
On the other hand, the polygonal-circle representations of these matrices
(Figures
8 and 10) are mutually symmetrical
too.

Matrices V4 and V5
when m = 5

Figure 11

The sum V4 + V5 is
a positive cycle matrix of period 1 (all cycles are constant), with two
axes of symmetry.

The six polygonal-circle representations of permutation
matrices in the case m=7

Figure 13

The nine polygonal-circle representations of permutation
matrices in the case m=10

Figure 14

We are now in a position to extrapolate and to formulate
a general conjecture. Its proof follows directly from the understanding
of the reflection of the branches of any alternating cycle in the sides
of the square circumscribed to the matrix. The theorem may be formulated
as follows:

Theorem 6

Let A and B be two equivalent positive alternating
cycle matrices of dimensions m × m, and let Vj be the
simultaneously permutation and positive alternating cycle matrix that transforms
A into B. Consider 2m dots equally spaced around a circle, numbered clockwise
1, 2, …, m, m, m-1, …, 2,
1, being the first m dots on the right side and the other m dots
on the left side. The polygonal-circle representation of Vj
is characterised by or may be described by the following construction steps:

(1) If j is an even number, start at the left
1-dot and advance clockwise each time j dots, until returning to
the left 1-dot and thus closing the polygonal line;

(2) If j is an odd number, start at the right 1-dot
and advance each time j-1 dots, until returning to the left 1-dot
and thus closing the polygonal line;

(3) If the closed polygonal line has m vertices,
stop. If the closed polygonal line has s vertices, where s is a divisor
of m (let us say m = st), then rotate the closed polygonal line about an
angle of (360o / t) around the circle centre and copy it. Repeat
this process t-1 times.

The resulting representation is a regular m-gon, a
regular star m-gon or a regular point star with m vertices. A point star
results in the case that m is an even number and j = m. The representations
of V2k and V2k+1
are mutually symmetric, where 2k+1 £
m.Similar results to the ones announced in theorems 4
to 6 can be proven for negative alternating cycle matrices.